Quantum and Semiquantum Pseudometrics and Applications
aa r X i v : . [ m a t h . A P ] F e b QUANTUM AND SEMIQUANTUM PSEUDOMETRICS ANDAPPLICATIONS
FRANC¸ OIS GOLSE AND THIERRY PAUL
Abstract.
We establish a Kantorovich duality for he pseudometric E ̵ h intro-duced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. (2017), 57–94],obtained from the usual Monge-Kantorovich distance dist MK , between classi-cal densities by quantization of one side of the two densities involved. We showseveral type of inequalities comparing dist MK , , E ̵ h and MK ̵ h , a full quantumanalogue of dist MK , introduced in [F. Golse, C. Mouhot, T. Paul, Commun.Math. Phys. (2016), 165–205], including an up to ̵ h triangle inequalityfor MK ̵ h . Finally, we show that, when nice optimal Kantorovich potentialsexist for E ̵ h , optimal couplings induce classical/quantum optimal transportsand the potentials are linked by a semiquantum Legendre type transform. Contents
1. Introduction and statement of some main results 12. Preliminaries 32.1. Monotone Convergence 32.2. Finite Energy Condition 52.3. Energy and Partial Trace 73. Couplings 84. Triangle Inequalities 105. Applications 186. Kantorovich duality for E ̵ h E ̵ h I: inequalities between
M K ̵ h , E ̵ h anddist MK , . 288. Applications of duality for E ̵ h II: “triangle” inequalities 289. Applications of duality for E ̵ h III: Classical/quantum optimal transportand semiquantum Legendre transform 309.1. A classical/quantum optimal transport 309.2. A semiquantum Legendre transform 32References 331.
Introduction and statement of some main results
The Monge-Kantorovich distance, also called Wasserstein distance, of exponenttwo on the phase-space T ∗ R d ∼ R d is defined, for two probability measures by(1) dist MK , ( µ, ν ) = inf π ∈ π [ µ.ν ] ∫ R d × R d (( q − q ′ ) + ( q − p ′ ) ) π ( dqdp, dq ′ dp ′ ) Date : February 11, 2021. where π [ µ, ν ] is the set of couplings π of µ, ν , i.e. the set of probability measures π on R d × R d such that for all test functions a, b ∈ C c ( R d ) we have that ∫ R d × R d ( a ( q, p ) + b ( q ′ , p ′ )) π ( dqdp, dq ′ dp ′ ) =∫ R d ( a ( q, p ) µ ( dqdp ) + b ( q ′ , p ′ ) ν ( dq ′ dp ′ )) . Among the many properties of dist MK , , let us mention the Kantorovich dualitywich stipulates that(2)dist MK , ( µ, ν ) = max a,b ∈ C b ( R d ) a ( q,p )+ b ( q ′ ,p ′ )≤( q − q ′ ) +( p − p ′ ) ∫ R d ( a ( q, p ) µ ( dqdp ) + b ( q, p ) ν ( dqdp ) , and the Knott-Smith-Brenier Theorem which says that, under certain conditionson µ, ν , any coupling π op satisfying(3) dist MK , ( µ, ν ) = ∫ R d × R d (( q − q ′ ) + ( q − p ′ ) ) π op ( dqdp, dq ′ dp ′ ) is supported in the graph of the convex function ( q + p ) − a op ( q, p ) where a op isan optimal function such that a op , b op provide the max in (2) for some b op .Finally, ( q + p ) − a op ( q, p ) and ( q + p ) − b op ( q, p ) are proven to be the Legendretransform of each other.A quantum version of dist MK , was proposed in [6] following the general rules ofquantization consisting in replacing ● probability measures µ.ν on phase-space T ∗ R d by quantum states R, S , i.e.density operators, i.e. positive trace one operators on L ( R d ) ● ∫ T ∗ R d by trace L ( R d ) ● couplings of µ, ν by density operators Π on L ( R d ) ⊗ L ( R d ) such that, forany bounded operators A, B , trace L ( R d )⊗ L ( R d ) ( A ⊗ I ) Π ) = trace L ( R d ) AS and trace L ( R d )⊗ L ( R d ) ( I ⊗ B ) Π ) = trace L ( R d ) BR . ● the cost function ( q − q ′ ) + ( q − p ′ ) by its Weyl pseudodifferential quanti-zation C = ( x − x ′ ) + ( − i ̵ h ∇ x + i ̵ h ∇ x ′ ) on L ( R d × R d ) .These considerations lead to the definition, for two density operators R, S on L ( R ) ,(4) M K ̵ h ( R, S ) = inf Π coupling R and S trace L ( R d )⊗ L ( R d ) C Π . The pseudometric
M K ̵ h has been extensively studied in [6], with applicationsto the study of the quantum mean-field limit uniformly in ̵ h , used in [2] for quan-tum optimal transport considerations and applied in [3] for the quantum bipartitematching problem. In particular, a Kantorovich duality was proven for M K ̵ h in [2]expressed as the following identity(5) M K ̵ h ( R, S ) = sup A = A ∗ ,B = B ∗ ∈ L( L ( R d )) A ⊗ I + I ⊗ B ≤ C trace ( AR + BS ) and the supremum was proven to be attended for two oparors ¯ A, ¯ B defined respec-tively on two Gelfand triplest surrounding L ( R d ) (see [2])Though M K ̵ h is symmetric in its argument, it is not a distance as one can easilyshow ([6]) that M K ̵ h ≥ d ̵ h . Nevertheless, one of the main result of this article UANTUM AND SEMIQUANTUM PSEUDOMETRICS 3 will be to prove the following (approximate) triangle inequality, valid for densityoperators
R, S, T (see Theorem 8.1 ( iii ) below)(6) M K ̵ h ( R, T ) ≤ M K ̵ h ( R, S ) + M K ̵ h ( S, T ) + d ̵ h. Actually, (6) is proved by using a kind of “semiquantum” generalisation ofdist MK , , defined in [7] and constructed by, roughly speaking, applying the quanti-zation rule aforementioned to only one on the two parts involved in dist MK , ( µ, ν ) :for f probability density on R d and R density operator on L ( R d ) we define(7) E ̵ h ( f, R ) = sup Π ( q,p ) density operatorssuch that trace Π ( q,p ) = f ( q,p ) and ∫ R d Π ( q,p ) = R ∫ R d trace L ( R d ,dx ) (( q − x ) + ( p + i ̵ h ∇ x ) ) Π ( q, p ) dqdp. The pseudometric E ̵ h has been used in [7] in order to derive several results concern-ing the quantum, uniform in ̵ h , mean-field derivation and in [7, 8] for semicalssicalpropagation estimates involving low regularity of the potential and the initial data(in particular with respect to the dimension, i.e. also to the number of particlespresent in the quantum evolution).In the present paper, we prove a Kantorovich duality for E ̵ h (Section 6, Theorem6.1), namely(8) E ̵ h ( f, R ) = sup a ∈ C b ( R d ) , B ∈ L( L ( R d )) a ( q,p )+ B ≤ ( q − x ) +( p + i ̵ h ∇ x ) ∫ R d a ( q, p ) f ( q, p ) dqdp + trace L ( R d ) BR, and then apply this duality to derive inequalities, such as (6), involving
M K ̵ h , E ̵ h and dist MK , , Theorems 7.1 and 8.1.In the last section of the paper, Section 9, we investigate the semiquantumanalogue of the Knott-Smith-Brenier Theorem and a semiquantum analogue of theLegendre transform: if E ̵ h ( f, R ) = ∫ R d a op ( q, p ) f ( q, p ) dqdp + trace L ( R d ) B op R, then a ( q, p ) ∶ = ( p + q − a op ( q, p )) is the semiquantum-Legendre transform of B ∶ = (−∇ x + x − B op ) , in the sense that a ( q, p ) = sup φ ∈ Dom ( B ) ( q ⋅ ⟨ φ ∣ x ∣ φ ⟩ + p ⋅ ⟨ φ ∣ − i ̵ h ∇ x ∣ φ ⟩ − ⟨ φ ∣ B ∣ φ ⟩) . Preliminaries
We have gathered together in this section some functional analytic remarks usedrepeatedly in the sequel.2.1.
Monotone Convergence.
We recall the analogue of the Beppo Levi mono-tone convergence theorem for operators in the form convenient for our purpose.Let H be a separable Hilbert space and 0 ≤ T = T ∗ ∈ L ( H ) . For each completeorthonormal system ( e j ) j ≥ of H , settrace H ( T ) = ∥ T ∥ ∶ = ∑ j ≥ ⟨ e j ∣ T ∣ e j ⟩ ∈ [ , +∞ ] . F. GOLSE AND T. PAUL
See Theorem 2.14 in [13]; in particular the expression on the last right hand side ofthese equalities is independent of the complete orthonormal system ( e j ) j ≥ . Then T ∈ L ( H ) ⇐⇒ ∥ T ∥ < ∞ . Lemma 2.1 (Monotone convergence) . Consider a sequence T n = T ∗ n ∈ L ( H ) suchthat ≤ T ≤ T ≤ . . . ≤ T n ≤ . . . , and sup n ≥ ⟨ x ∣ T n ∣ x ⟩ < ∞ for all x ∈ H . Then(a) there exists T = T ∗ ∈ L ( H ) such that T n → T weakly as n → ∞ , and(b) trace H ( T n ) → trace H ( T ) as n → ∞ .Proof. Since the sequence ⟨ x ∣ T n ∣ x ⟩ ∈ [ , +∞ ) is nondecreasing for each x ∈ H , ⟨ x ∣ T n ∣ x ⟩ → sup n ≥ ⟨ x ∣ T n ∣ x ⟩ = ∶ q ( x ) ∈ [ , +∞ ) for all x ∈ H as n → ∞ . Hence ⟨ x ∣ T n ∣ y ⟩ = ⟨ y ∣ T n ∣ x ⟩ → ( q ( x + y ) − q ( x − y ) + iq ( x − iy ) − iq ( x + iy )) = ∶ b ( x, y ) ∈ C as n → +∞ . By construction, b is a nonnegative sesquilinear form on H .Consider, for each k ≥ F k ∶ = { x ∈ H s.t. ⟨ x ∣ T n ∣ x ⟩ ≤ k for each n ≥ } . The set F k is closed for each k ≥
0, being the intersection of the closed sets definedby the inequality ⟨ x ∣ T n ∣ x ⟩ ≤ k as n ≥
1. Since the sequence ⟨ x ∣ T n ∣ x ⟩ is bounded foreach x ∈ H , ⋃ k ≥ F k = H . Applying Baire’s theorem shows that there exists N ≥ F N / = ∅ . In otherwords, there exists r > x ∈ H such that ∣ x − x ∣ ≤ r Ô⇒ ∣⟨ x ∣ T n ∣ x ⟩∣ ≤ N for all n ≥ . By linearity and positivity of T n , this implies ∣⟨ z ∣ T n ∣ z ⟩∣ ≤ r ( M + N )∥ z ∥ for all n ≥ , with M ∶ = sup n ≥ ⟨ x ∣ T n ∣ x ⟩ . In particularsup ∣ z ∣ ≤ q ( z ) ≤ r ( M + N ) , so that ∣ b ( x, y )∣ ≤ r ( M + N )∣∥ x ∥ H ∥ y ∥ H for each x, y ∈ H by the Cauchy-Schwarz inequality. By the Riesz representationtheorem, there exists T ∈ L ( H ) such that T = T ∗ ≥ , and b ( x, y ) = ⟨ x ∣ T ∣ y ⟩ . This proves (a). Observe that T ≥ T n for each n ≥
1, so thatsup n ≥ trace H ( T n ) ≤ trace H ( T ) . In particular sup n ≥ trace H ( T n ) = +∞ Ô⇒ trace H ( T ) = +∞ . Since the sequence trace H ( T n ) is nondecreasing,trace H ( T n ) → sup n ≥ trace H ( T n ) as n → ∞ . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 5
By the noncommutative variant of Fatou’s lemma (Theorem 2.7 (d) in [13]),sup n ≥ trace H ( T n ) < ∞ Ô⇒ T ∈ L ( H ) and trace H ( T ) ≤ sup n ≥ trace H ( T n ) . Since the opposite inequality is already known to hold, this proves (b). (cid:3)
Here is a convenient variant of this lemma.
Corollary 2.2.
Consider a sequence T n = T ∗ n ∈ L ( H ) such that ≤ T ≤ T ≤ . . . ≤ T n ≤ . . . , and sup n ≥ trace H ( T n ) < ∞ . Then there exists T ∈ L ( H ) such that T n → T weakly as n → ∞ , and T = T ∗ ≥ , and trace H ( T ) = lim n → ∞ trace H ( T n ) . Proof.
Since any x ∈ H ∖ { } can be normalized and completed into a completeorthonormal system of H , one hassup n ≥ ⟨ x ∣ T ∣ x ⟩ ≤ ∥ x ∥ H sup n ≥ trace H ( T n ) < ∞ . One concludes by applying Lemma 2.1 (a) and (b). (cid:3)
Finite Energy Condition.
In the sequel, we shall repeatedly encounter thefollowing typical situation. Let A = A ∗ ≥ H with domain Dom ( A ) , and let E be its spectral decomposition.Let T ∈ L ( H ) satisfy T = T ∗ ≥
0, and let ( e j ) j ≥ be a complete orthonormalsystem of eigenvectors of T with T e j = τ j e j and τ j ∈ [ , +∞ ) for each j ≥ Lemma 2.3.
Assume that (9) ∑ j ≥ τ j ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e j ⟩ < ∞ . Then T / AT / ∶ = ∑ j,k ≥ τ / j τ / k ( ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e k ⟩) ∣ e j ⟩⟨ e k ∣ satisfies ≤ T / AT / = ( T / AT / ) ∗ ∈ L ( H ) and trace H ( T / AT / ) = ∑ j ≥ τ j ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e j ⟩ . Proof.
For each Borel ω ⊂ R and each x, y ∈ H , one has ∣⟨ x ∣ E ( ω )∣ y ⟩∣ = ∣⟨ E ( ω ) x ∣ E ( ω ) y ⟩∣ ≤ ∥ E ( ω ) x ∥∥ E ( ω ) y ∥ = ⟨ x ∣ E ( ω )∣ x ⟩ / ⟨ y ∣ E ( ω )∣ y ⟩ / since E ( ω ) is a self-adjoint projection. In particular, for each α >
0, one has2 ∣⟨ x ∣ E ( ω )∣ y ⟩∣ ≤ α ⟨ x ∣ E ( ω )∣ x ⟩ + α ⟨ y ∣ E ( ω )∣ y ⟩ . Hence a jk ∶ = ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e k ⟩ ∈ C and satisfies 2 ∣ a jk ∣ ≤ αa jj + α a kk for all α >
0, so that ∣ a jk ∣ ≤ a jj a kk F. GOLSE AND T. PAUL for all j, k ≥ ( τ j a jj ) j ≥ ∈ ℓ ( N ∗ ) by (9) and since ⟨ e j ∣ T / AT / ∣ e k ⟩ = τ / j τ / k a jk = ⟨ e k ∣ T / AT / ∣ e j ⟩ , one concludes that T / AT / = ( T / AT / ) ∗ ∈ L ( H ) . Moreover, for each x ∈ H ⟨ x ∣ T / AT / ∣ x ⟩ = ∑ j,k ≥ τ / j τ / k ⟨ e j ∣ x ⟩⟨ e k ∣ x ⟩ ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e k ⟩ ≥ ∫ ∞ λ ⟨ ∑ j ≥ τ / j ⟨ e j ∣ x ⟩ e j ∣ E ( dλ )∣ ∑ j ≥ τ / j ⟨ e j ∣ x ⟩ e j ⟩ = ∫ ∞ λ ⟨ T / x ∣ E ( dλ )∣ T / x ⟩ ≥ , so that T / AT / ≥ ∑ l ≥ ⟨ e l ∣ T / AT / ∣ e l ⟩ = ∑ l ≥ ∑ j,k ≥ τ / j τ / k ( ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e k ⟩)⟩ e l ∣ e j ⟩⟨ e k ∣ e l ⟩ = ∑ l ≥ ∑ j,k ≥ τ / j τ / k ( ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e k ⟩) δ lj δ lk = ∑ l ≥ τ l ∫ ∞ λ ⟨ e l ∣ E ( dλ )∣ e l ⟩ < ∞ so that ∥ T / AT / ∥ = trace H ( T / AT / ) = ∑ l ≥ τ l ∫ ∞ λ ⟨ e l ∣ E ( dλ )∣ e l ⟩ < ∞ and in particular T / AT / ∈ L ( H ) . (cid:3) Corollary 2.4.
Let T ∈ L ( H ) satisfy T = T ∗ ≥ and (9) . Let Φ n ∶ R + → R + be asequence of continuous, bounded and nondecreasing functions such that ≤ Φ ( r ) ≤ Φ ( r ) ≤ . . . ≤ Φ n ( r ) → r as n → ∞ . Set Φ n ( A ) ∶ = ∫ ∞ Φ n ( λ ) E ( dλ ) ∈ L ( H ) . Then Φ n ( A ) = Φ n ( A ) ∗ ≥ for each n ≥ and, for each T ∈ L ( H ) such that T = T ∗ ≥ , the sequence T / Φ n ( A ) T / converges weakly to T / AT / as n → ∞ .Moreover trace H ( T Φ n ( A )) → trace H ( T / AT / ) as n → ∞ . Proof.
Since E is a resolution of the identity on [ , +∞ ) , and since Φ n is continuous,bounded and with values in [ , +∞ ) , the operators Φ n ( A ) satisfy0 ≤ Φ n ( A ) = Φ n ( A ) ∗ ≤ ( sup z ≥ Φ n ( z )) I H and 0 ≤ Φ ( A ) ≤ Φ ( A ) ≤ . . . ≤ Φ n ( A ) ≤ . . . Set R n ∶ = T / Φ n ( A ) T / ; by definition 0 ≤ R n = R ∗ n ∈ L ( H ) and one has0 ≤ R ≤ R ≤ . . . ≤ R n ≤ . . . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 7 together withtrace H ( R n ) = ∑ j ≥ τ j ∫ ∞ Φ n ( λ )⟨ e j ∣ E ( dλ )∣ e j ⟩ ≤ ∑ j ≥ τ j ∫ ∞ λ ⟨ e j ∣ E ( dλ )∣ e j ⟩ < ∞ by (9). Applying Corollary 2.2 shows that R n converges weakly to some R ∈ L ( H ) such that R = R ∗ ≥
0. Finally T / AT / − R n = ∑ j,k ≥ τ / j τ / k ( ∫ ∞ ( λ − Φ n ( λ ))⟨ e j ∣ E ( dλ )∣ e k ⟩) ∣ e j ⟩⟨ e k ∣ so that ⟨ x ∣ T / AT / − R n ∣ x ⟩ = ∫ ∞ ( λ − Φ n ( λ ))⟨ ∑ j ≥ τ / j ⟨ e j ∣ x ⟩ e j ∣ E ( dλ )∣ ∑ k ≥ τ / k ⟨ e k ∣ x ⟩ e k ⟩ = ∫ ∞ ( λ − Φ n ( λ ))⟨ T / x ∣ E ( dλ )∣ T / x ⟩ ≥ . Hence 0 ≤ T / AT / − R n = ( T / AT / − R n ) ∗ ∈ L ( H ) so that ∥ T / AT / − R n ∥ = trace H ( T / AT / − R n ) = ∑ j ≥ τ j ∫ ∞ ( λ − Φ n ( λ ))⟨ e j ∣ E ( dλ )∣ e j ⟩ → n → ∞ by monotone convergence. Hence R n → T / AT / in L ( H ) and one hasin particulartrace H ( T Φ n ( A )) = trace H ( T / Φ n ( A ) T / ) → trace H ( T / AT / ) . (cid:3) Energy and Partial Trace.
Let H and H be two separable Hilbert spaces.Let A = A ∗ ≥ H with domain Dom ( A ) ,and let E be its spectral decomposition. Let S ∈ L ( H ) satisfy S = S ∗ ≥
0, andlet ( e j ) j ≥ be a complete orthonormal system of H of eigenvectors of S , witheigenvalues ( σ j ) j ≥ such that Se j = σ j e j for each j ≥
1. Assume that ∑ j ≥ σ j ∫ +∞ λ ⟨ e j ∣ E ( dλ )∣ e j ⟩ < ∞ . Lemma 2.5.
Let T ∈ L ( H ⊗ H ) satisfy the partial trace condition trace ( T ∣ H ) = S .
Then T / ( A ⊗ I H ) T / ∈ L ( H ⊗ H ) and trace H ⊗H ( T / ( A ⊗ I ) T / ) = trace H ( S / AS / ) . Proof.
For all n ≥
1, set A n = Φ n ( A ) ∈ L ( H ) , withΦ n ( r ) ∶ = r + n r , for all r ≥ . By construction, one has A n = A ∗ n ≥ A ≤ A ≤ . . . ≤ A n ≤ . . . F. GOLSE AND T. PAUL
Hence T / ( A n ⊗ I H ) T / = ( T / ( A n ⊗ I H ) T / ) ∗ ≥ n ≥
1, and T / ( A ⊗ I H ) T / ≤ T / ( A ⊗ I H ) T / ≤ . . . ≤ T / ( A n ⊗ I H ) T / ≤ . . . and sincetrace H ⊗H ( T ( A n ⊗ I H )) = trace H ( SA n ) → trace H ( S / AS / ) as n → ∞ by the partial trace condition and Corollary 2.4, we conclude fromCorollary 2.2 that T / ( A ⊗ I H ) T / = ( T / ( A ⊗ I H ) T / ) ∗ ≥ H ⊗H ( T / ( A ⊗ I ) T / ) = trace H ( S / AS / ) . (cid:3) Couplings
Let H ∶ = L ( R d ) . An operator R ∈ L ( H ) is a density operator if R = R ∗ ≥ ( R ) = . We denote by D ( H ) the set of density operators on H , and define D ( H ) ∶ = { R ∈ D ( H ) s.t. trace ( R / (∣ y ∣ − ∆ y ) R / ) < ∞ } . The set of Borel probability measures on R d × R d is denoted by P ( R d × R d ) . Wedenote by P ( R d × R d ) the set of Borel probability measures µ on R d × R d suchthat ∬ R d × R d (∣ x ∣ + ∣ ξ ∣ ) µ ( dxdξ ) < ∞ . The set of Borel probability measures on R d × R d which are absolutely continuouswith respect to the Lebesgue measure on R d × R d is denoted P ac ( R d × R d ) ⊂P ( R d × R d ) . We set P ac ( R d × R d ) = P ac ( R d × R d ) ∩ P ( R d × R d ) , and we identifyelements of P ac ( R d × R d ) with their densities with respect to the Lebesgue measure.Let R , R ∈ D ( H ) ; a coupling of R and R is an element R ∈ D ( H ⊗ H ) suchthat trace H ⊗ H (( A ⊗ I + I ⊗ B ) R ) = trace H ( R A ) + trace H ( R B ) . The set of couplings of R and R will be denoted by C ( R , R ) . Obviously thetensor product R ⊗ R ∈ C ( R , R ) , so that C ( R , R ) / = ∅ .Let f be a probability density on R d × R d , and let R ∈ D ( H ) . A coupling of f and R is an ultraweakly measurable operator-valued function ( x, ξ ) ↦ Q ( x, ξ ) defined a.e. on R d × R d with values in L ( H ) such that Q ( x, ξ ) = Q ( x, ξ ) ∗ ≥ , ∬ R d × R d Q ( x, ξ ) dxdξ = R and trace H ( Q ( x, ξ )) = f ( x, ξ ) ˆE for a.e. ( x, ξ ) ∈ R d × R d . The set of couplings of f and R will also be denoted by C ( f, R ) . Since the map ( x, ξ ) ↦ f ( x, ξ ) R (henceforth denoted f ⊗ R ) obviously belongs to C ( f, R ) , one has C ( f, R ) / = ∅ .In general, one does not know much about the general structure of couplings be-tween two density operators. However, the case where one of the density operatorsis a rank 1 projection is particularly simple. UANTUM AND SEMIQUANTUM PSEUDOMETRICS 9
Lemma 3.1.
Let P = P ∗ ∈ L ( H ) be a rank projection. Then(i) for each probability density f on R d × R d , one has C ( f, P ) = { f ⊗ P } ;(ii) for each R ∈ D ( H ) , one has C ( P, R ) = { P ⊗ R } and C ( R, P ) = { R ⊗ P } . This is in complete analogy with the following elementary observation: if µ ∈ P ( R d ) and y ∈ R d , the only coupling of µ and δ y is µ ⊗ δ y . In other words,self-adjoint rank-1 projections are the quantum analogue of points in this picture. Proof.
Let Q ∈ C ( f, P ) ; one has ∬ R d × R d trace H (( I − P ) Q ( x, ξ )( I − P )) dxdξ = trace H (( I − P ) ∬ R d × R d Q ( x, ξ ) dxdξ ( I − P ) = trace H (( I − P ) P ( I − P )) = . Since ( I − P ) Q ( x, ξ )( I − P ) ≥ ( x, ξ ) ∈ R d × R d , this implies that ( I − P ) Q ( x, ξ )( I − P ) = ( x, ξ ) ∈ R d × R d . Since Q ( x, ξ ) = Q ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d × R d , we deduce from the Cauchy-Schwarz inequality that, for all φ, ψ ∈ H ∣⟨ P φ ∣ Q ( x, ξ )∣( I − P ) ψ ⟩∣ = ∣⟨( I − P ) ψ ∣ Q ( x, ξ )∣ P φ ⟩∣ ≤ ⟨ P φ ∣ Q ( x, ξ )∣ P φ ⟩ / ⟨( I − P ) ψ ∣ Q ( x, ξ )∣( I − P ) ψ ⟩ / = . Hence ( I − P ) Q ( x, ξ ) P = P Q ( x, ξ )( I − P ) = ( x, ξ ) ∈ R d × R d , so that Q ( x, ξ ) = P Q ( x, ξ ) P for a.e. ( x, ξ ) ∈ R d × R d . Writing P as P = ∣ u ⟩⟨ u ∣ where u ∈ H is a unit vector, we conclude that Q ( x, ξ ) = ⟨ u ∣ Q ( x, ξ )∣ u ⟩ P for a.e. ( x, ξ ) ∈ R d × R d . Finally trace ( Q ( x, ξ )) = f ( x, ξ ) = ⟨ u ∣ Q ( x, ξ )∣ u ⟩ for a.e. ( x, ξ ) ∈ R d × R d . This concludes the proof of (i).As for (ii), let Q ∈ C ( R, P ) . Thentrace H ⊗ H (( I ⊗ ( I − P )) Q ( I ⊗ ( I − P ))) = trace H ( I − P ) P ( I − P )) = . Hence ( I ⊗ ( I − P )) Q ( I ⊗ ( I − P ) = . Since Q = Q ∗ ≥
0, the Cauchy-Schwarz inequality implies that, for all φ, φ ′ , ψ, ψ ′ ∈ H ∣⟨ φ ⊗ ψ ∣( I ⊗ P ) Q ( I ⊗ ( I − P ))∣ φ ′ ⊗ ψ ′ ⟩∣ = ∣⟨ φ ′ ⊗ ψ ∣ ′ ( I ⊗ ( I − P )) Q ( I ⊗ P )∣ φ ⊗ ψ ⟩∣ ≤ ⟨ φ ⊗ ψ ∣( I ⊗ P ) Q ( I ⊗ P )∣ φ ⊗ ψ ⟩ / ⟨ φ ′ ⊗ ψ ′ ∣( I ⊗ ( I − P )) Q ( I ⊗ ( I − P ))∣ φ ′ ⊗ ψ ′ ⟩ / so that ( I ⊗ P ) Q ( I ⊗ ( I − P )) = ( I ⊗ ( I − P )) Q ( I ⊗ P ) = . Hence Q = ( I ⊗ P ) Q ( I ⊗ P ) . Writing P = ∣ u ⟩⟨ u ∣ with u ∈ H and ∣ u ∣ = ⟨ φ ⊗ ψ ∣ Q ∣ φ ′ ⊗ ψ ′ ⟩ = ⟨ φ ⊗ u ∣ Q ∣ φ ′ ⊗ u ⟩⟨ u ∣ ψ ⟩⟨ u ∣ ψ ′ ⟩ . This shows that Q = L ⊗ ∣ u ⟩⟨ u ∣ = L ⊗ P , where L = L ∗ is the element of L ( H ) suchthat ⟨ φ ∣ L ∣ φ ′ ⟩ = ⟨ φ ⊗ u ∣ Q ∣ φ ′ ⊗ u ⟩ for each φ, φ ′ ∈ H . (Observe indeed that ( φ, φ ′ ) ↦ ⟨ φ ⊗ u ∣ Q ∣ φ ′ ⊗ u ⟩ is a continuous,symmetric bilinear functional on H , and is therefore represented by a unique self-adjoint element of L ( H ) .) We conclude by observing thattrace H ⊗ H (( A ⊗ I ) Q ) = trace H ( AR ) = trace H ( LR ) for each finite rank operator A ∈ L ( H ) , and this implies that Q = R ⊗ P .The case of Q ′ ∈ C ( P, R ) is handled similarly. (cid:3) Next we explain how to “disintegrate” a coupling with respect to one of itsmarginals when this marginal is a probability density.
Lemma 3.2.
Let f ∈ P ac ( R d × R d ) , let R ∈ D ( H ) and let Q ∈ C ( f, R ) . Thereexists a σ ( L ( H ) , L ( H )) weakly measurable function ( x, ξ ) ↦ Q f ( x, ξ ) defined a.e.on R d × R d with values in L ( H ) such that Q f ( x, ξ ) = Q ∗ f ( x, ξ ) ≥ , trace ( Q f ( x, ξ )) = , and Q ( x, ξ ) = f ( x, ξ ) Q f ( x, ξ ) for a.e. ( x, ξ ) ∈ R d × R d .Proof. Let f be a Borel measurable function defined on R d × R d and such that f ( x, ξ ) = f ( x, ξ ) for a.e. ( x, ξ ) ∈ R d × R d . Let N be the Borel measurable setdefined as follows: N ∶ = {( x, ξ ) ∈ R d × R d s.t. f ( x, ξ ) = } , and let u ∈ H satisfy ∣ u ∣ =
1. Consider the function ( x, ξ ) ↦ Q f ( x, ξ ) ∶ = Q ( x, ξ ) + N ( x, ξ )∣ u ⟩⟨ u ∣ f ( x, ξ ) + N ( x, ξ ) ∈ L ( H ) defined a.e. on R d × R d . The function f + N > R d × R d while ( x, ξ ) ↦ ⟨ φ ∣ Q ( x, ξ )∣ ψ ⟩ is measurable and defined a.e. on R d × R d for each φ, ψ ∈ H . Set A ∶ L ( H ) × ( , +∞ ) ∋ ( T, λ ) ↦ λ − T ∈ L ( H ) ; since A is continuous, thefunction Q f ∶ = A ( Q + N ⊗ ∣ u ⟩⟨ u ∣ , f + N ) is weakly measurable on R d × R d . Since f + N >
0, and since Q ( x, ξ ) = Q ∗ ( x, ξ ) ≥
0, one has ( Q ( x, ξ ) + N ⊗ ∣ u ⟩⟨ u ∣) ∗ = Q ( x, ξ ) + N ⊗ ∣ u ⟩⟨ u ∣ ≥ ( x, ξ ) ∈ R d × R d . On the other hand, for a.e. ( x, ξ ) ∈ R d × R d , one has trace ( Q ( x, ξ ) + N ⊗ ∣ u ⟩⟨ u ∣) = f ( x, ξ ) + N ( x, ξ ) , so thattrace ( Q f ( x, ξ )) =
1. Finally f ( x, ξ ) Q f ( x, ξ ) = f ( x, ξ ) Q ( x, ξ ) f ( x, ξ ) + N ( x, ξ ) = Q ( x, ξ ) for a.e. ( x, ξ ) ∈ R d × R d , since f = f a.e. on R d × R d and N ( x, ξ ) = ( x, ξ ) ∈ R d × R d such that f ( x, ξ ) >
0. Since Q f satisfies trace ( Q f ( x, ξ )) = ( x, ξ ) ∈ R d × R d and isweakly measurable on R d × R d , it is σ ( L ( H ) , L ( H )) weakly measurable. (cid:3) Triangle Inequalities
The following “pseudo metrics” have been defined in [6] and in [7] respectively.
Definition 4.1.
For all
R, S ∈ D ( H ) and all f ∈ P ac ( R d × R d ) , we set M K ̵ h ( R, S ) ∶ = inf A ∈ C( R,S ) trace H ⊗ H ( A / CA / ) / where C ∶ = C ( x, y, ̵ hD x , ̵ hD y ) = ∣ x − y ∣ + ∣̵ hD x − ̵ hD y ∣ . Similarly, we set E ̵ h ( f, R ) ∶ = inf a ∈ C( f,R ) ( ∬ R d × R d trace H ( a ( x, ξ ) / c ( x, ξ ) a ( x, ξ ) / ) dxdξ ) / where c ( x, ξ ) ∶ = c ( x, ξ, y, ̵ hD y ) = ∣ x − y ∣ + ∣ ξ − ̵ hD y ∣ . The above “pseudometrics” satisfy the following inequalities.
Theorem 4.2.
Let f, g ∈ P ac ( R d × R d ) , and let R , R , R ∈ D ( H ) . The followinginequalities hold true:(a) E ̵ h ( f, R ) ≤ dist MK , ( f, g ) + E ̵ h ( g, R ) ;(b) M K ̵ h ( R , R ) ≤ E ̵ h ( f, R ) + E ̵ h ( f, R ) ;(c) if rank ( R ) = , then M K ̵ h ( R , R ) ≤ M K ̵ h ( R , R ) + M K ̵ h ( R , R ) , (d) if rank ( R ) = , then dist MK , ( f, g ) ≤ E ̵ h ( f, R ) + E ̵ h ( g, R ) , (e) if rank ( R ) = , then E ̵ h ( f, R ) ≤ E ̵ h ( f, R ) + M K ̵ h ( R , R ) . The proofs of all these triangle inequalities make use of some inequalities betweenthe (classical and/or quantum) transportation cost operators. We begin with anelementary, but useful lemma, which can be viewed as the Peter-Paul inequality foroperators.
Lemma 4.3.
Let
T, S be unbounded self-adjoint operators on H = L ( R n ) , withdomains Dom ( T ) and Dom ( S ) respectively such that Dom ( T ) ∩ Dom ( S ) is densein H . Then, for all α > , one has ⟨ v ∣ T S + ST ∣ v ⟩ ≤ α ⟨ v ∣ T ∣ v ⟩ + α ⟨ v ∣ S ∣ v ⟩ , for all v ∈ Dom ( T ) ∩ Dom ( S ) . Proof.
Indeed, for each α > v ∈ Dom ( T ) ∩ Dom ( S ) , one has α ⟨ v ∣ T ∣ v ⟩ + α ⟨ v ∣ S ∣ v ⟩ − ⟨ v ∣ T S + ST ∣ v ⟩ = ∣√ αT v ∣ + ∣ √ α Sv ∣ − ⟨√ αT v ∣ √ α Sv ⟩ − ⟨ √ α Sv ∣√ αT v ⟩ = ∣√ αT v − √ α Sv ∣ ≥ . (cid:3) Lemma 4.4.
For each x, ξ, y, η, z ∈ R d and each α > , one has c ( x, ξ ; z, ̵ hD z ) ≤ ( + α )(∣ x − y ∣ + ∣ ξ − η ∣ ) + ( + α ) c ( y, η ; z, ̵ hD z ) ,C ( x, z, ̵ hD x , ̵ hD z ) ≤ ( + α ) c ( y, η ; x, ̵ hD x ) + ( + α ) c ( y, η ; z, ̵ hD z ) ,C ( x, z, ̵ hD x , ̵ hD z ) ≤ ( + α ) C ( x, y ; ̵ hD x , ̵ hD y ) + ( + α ) C ( y, z ; ̵ hD y , ̵ hD z ) , ∣ x − z ∣ + ∣ ξ − ζ ∣ ≤ ( + α ) c ( x, ξ ; y, ̵ hD y ) + ( + α ) c ( z, ζ ; y, ̵ hD y ) ,c ( x, ξ ; z, ̵ hD z ) ≤ ( + α ) c ( x, ξ ; y, ̵ hD y ) + ( + α ) C ( y, z ; ̵ hD y , ̵ hD z ) . All these inequalities are of the form A ≤ B where A and B are unboundedself-adjoint operators on L ( R n ) for some n ≥
1, with W ∶ = { ψ ∈ H ( R n ) s.t. ∣ x ∣ ψ ∈ H } ⊂ Dom f ( A ) ∩ Dom f ( B ) , denoting by Dom f ( A ) (resp. Dom f ( B ) ) the form-domain of A (resp. of B ) — see § VIII.6 in [9] on pp. 276–277. The inequality A ≤ B means that the bilinear formassociated to B − A is nonnegative, i.e. that ⟨ w ∣ A ∣ w ⟩ ≤ ⟨ w ∣ B ∣ w ⟩ , for all w ∈ W . Proof.
All these inequalities are proved in the same way. Let us prove for instancethe third inequality: C ( x, z, ̵ hD x , ̵ hD z ) = ∣ x − y + y − z ∣ + ∣̵ hD x − ̵ hD y + ̵ hD y − ̵ hD z ∣ = C ( x, y ; ̵ hD x , ̵ hD y ) + C ( y, z ; ̵ hD y , ̵ hD z ) + ( x − y ) ⋅ ( y − z ) + (̵ hD x − ̵ hD y ) ⋅ (̵ hD y − ̵ hD z ) . Observe indeed that the multiplication operators by ( x − y ) and by ( y − z ) commute;likewise (̵ hD x − ̵ hD y ) and (̵ hD y − ̵ hD z ) commute. By Lemma 4.32 ( x − y ) ⋅ ( y − z ) + (̵ hD x − ̵ hD y ) ⋅ (̵ hD y − ̵ hD z ) ≤ αC ( x, y ; ̵ hD x , ̵ hD y ) + α C ( y, z ; ̵ hD y , ̵ hD z ) , which concludes the proof of the third inequality. (cid:3) Proof of Theorem 4.2 (a).
By Theorem 2.12 in chapter 2 of [14], there exists anoptimal coupling for W ( f, g ) , of the form f ( x, ξ ) δ ∇ Φ ( x,ξ ) ( dydη ) , where Φ is aconvex function on R d × R d . Let Q ∈ C ( g, R ) and set P ( x, ξ ; dydη ) ∶ = f ( x, ξ ) δ ∇ Φ ( x,ξ ) ( dydη ) Q g ( y, η ) , where Q g is the disintegration of Q with respect to f obtained in Lemma 3.2. Then P is a nonnegative,self-adjoint operator-valued measure satisfyingtrace H ( P ( x, ξ ; dydη )) = f ( x, ξ ) δ ∇ Φ ( x,ξ ) ( dydη ) while ∫ P dxdξ = ( ∇ Φ f )( y, η ) dydηQ g ( y, η ) = g ( y, η ) Q g ( y, η ) dydη = Q ( y, η ) dydη . In particular(10) ∫ P ( x, ξ ; dydη ) = f ( x, ξ ) Q g ( ∇ Φ ( x, ξ )) ∈ C ( f, R ) . Therefore E ̵ h ( f, R ) ≤ ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( x, ξ ) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ . By the first inequality in Lemma 4.4, one has c ̵ h ( x, ξ ; z, ̵ hD z ) ≤ ( + α )∣( x, ξ ) − ∇ φ ( x, ξ )∣ + ( + α ) c ̵ h ( ∇ Φ ( x, ξ ) ; z, ̵ hD z ) for a.e. ( x, ξ ) ∈ R d × R d and all α >
0. Since g ∈ P ac ( R d × R d ) and R ∈ D ( H ) and Q ∈ C ( g, R ) , then ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ) Q ( y, η ) / ) dydη = ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( ∇ Φ ( x, ξ )) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ < ∞ . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 13
For each ǫ >
0, set c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) = ( I + ǫc ̵ h ( x, ξ ; z, ̵ hD z )) − c ̵ h ( x, ξ ; z, ̵ hD z ) ≤ c ̵ h ( x, ξ ; z, ̵ hD z ) . Then, for a.e. ( x, ξ ) ∈ R d × R d and each ǫ >
0, one has Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( ∇ Φ ( x, ξ ) ; z, ̵ hD z ) Q g ( ∇ Φ ( x, ξ )) / ∈ L ( H ) , and Q g ( ∇ Φ ( x, ξ )) / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) Q g ( ∇ Φ ( x, ξ )) / ≤ ( + α )∣( x, ξ ) − ∇ φ ( x, ξ )∣ Q g ( ∇ Φ ( x, ξ )) + ( + α ) Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( ∇ Φ ( x, ξ ) ; z, ̵ hD z ) Q g ( ∇ Φ ( x, ξ )) / . Integrating both sides of this inequality with respect to the probability distribution f ( x, ξ ) , one finds ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ǫ ̵ h ( x, ξ ) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ ≤ ( + α ) ∫ ∣( x, ξ ) − ∇ φ ( x, ξ )∣ f ( x, ξ ) dxdξ + ( + α ) ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( ∇ Φ ( x, ξ )) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ ≤ ( + α ) dist MK , ( f, g ) + ( + α ) ∫ trace H ( Q g ( y, η ) / c ̵ h ( y, η ) Q g ( y, η ) / ) g ( y, η ) dydη ≤ ( + α ) dist MK , ( f, g ) + ( + α ) ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ) Q ( y, η ) / ) dydη . Minimizing the last right hand side of this inequality in Q ∈ C ( g, R ) shows that ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ǫ ̵ h ( x, ξ ) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ ≤ ( + α ) dist MK , ( f, g ) + ( + α ) E ( g, R ) . Passing to the limit as ǫ → + in the left hand side and applying Corollary 2.4 showsthat E ̵ h ( f, R ) ≤ ∫ trace H ( Q g ( ∇ Φ ( x, ξ )) / c ̵ h ( x, ξ ) Q g ( ∇ Φ ( x, ξ )) / ) f ( x, ξ ) dxdξ ≤ ( + α ) dist MK , ( f, g ) + ( + α ) E ( g, R ) , the first inequality being a consequence of the definition of E ̵ h according to (10).Finally, minimizing the right hand side of this inequality as α >
0, i.e. choosing α = E ̵ h ( f, g )/ dist MK , ( f, g ) if f / = g a.e. on R d × R d , or letting α → +∞ if f = g , wearrive at the inequality E ̵ h ( f, R ) ≤ dist MK , ( f, g ) + E ̵ h ( g, R ) + E ̵ h ( g, R ) dist MK , ( f, g ) = ( dist MK , ( f, g ) + E ̵ h ( g, R )) , which is precisely the inequality (a). (cid:3) Proof of Theorem 4.2 (b).
Let Q ∈ C ( f, R ) and Q ∈ C ( f, R ) . Let Q ,f and Q ,f be the disintegrations of Q and Q with respect to f obtained in Lemma 3.2. Foreach ǫ >
0, set(11) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z ) = ( I + ǫC ̵ h ( x, z, ̵ hD x , ̵ hD z )) − C ̵ h ( x, z, ̵ hD x , ̵ hD z ) and observe that0 ≤ C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z ) = C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z ) ∗ ∈ L ( H ⊗ H ) . By the second inequality in Lemma 4.4, for all ( y, η ) ∈ R d × R d and all α >
0, onehas C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z ) ≤ C ̵ h ( x, z, ̵ hD x , ̵ hD z ) ≤ ( + α ) c ̵ h ( y, η ; x, ̵ hD x ) + ( + α ) c ̵ h ( y, η ; z, ̵ hD z ) . Therefore, for a.e. ( y, η ) ∈ R d × R d , one has ( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / ≤ ( + α )( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / c ̵ h ( y, η ; x, ̵ hD x )( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / + ( + α )( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / c ̵ h ( y, η ; z, ̵ hD z )( Q ,f ( y, η ) ⊗ Q ,f ( y, η )) / = ( + α ) ( Q ,f ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ,f ( y, η ) / ) ⊗ Q ,f ( y, η ) + ( + α ) Q ,f ( y, η ) ⊗ ( Q ,f ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ,f ( y, η ) / ) . Taking the trace in H ⊗ H of both sides of this inequality shows thattrace H ⊗ H (( Q ,f ⊗ Q ,f ( y, η )) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) = trace H ⊗ H (( Q ,f ⊗ Q ,f ( y, η )) / C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )( Q ,f ⊗ Q ,f ( y, η )) / ) ≤ ( + α ) trace H ( Q ,f ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ,f ( y, η ) / ) + ( + α ) trace H ( Q ,f ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ,f ( y, η ) / ) for a.e. ( y, η ) ∈ R d × R d . Integrating both sides of this inequality in ( y, η ) withrespect to f shows thattrace H ⊗ H (( ∫ ( Q ,f ⊗ Q ,f ( y, η )) f (( y, η ) dydη ) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) ≤ ( + α ) ∫ trace H ( Q ,f ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ,f ( y, η ) / ) f ( y, η ) dydη + ( + α ) ∫ trace H ( Q ,f ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ,f ( y, η ) / ) f ( y, η ) dydη = ( + α ) ∫ trace H (( f Q ,f ( y, η )) / c ̵ h ( y, η ; x, ̵ hD x )( f Q ,f ( y, η )) / ) dydη + ( + α ) ∫ trace H (( f Q ,f ( y, η )) / c ̵ h ( y, η ; z, ̵ hD z )( f Q ,f ( y, η )) / ) dydη = ( + α ) ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ( y, η ) / ) dydη + ( + α ) ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ( y, η ) / ) dydη . By construction P ∶ = ∫ ( Q ,f ⊗ Q ,f ( y, η )) f (( y, η ) dydη ∈ C ( R , R ) ;on the other hand ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ( y, η ) / ) dydη < ∞ ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ( y, η ) / ) dydη < ∞ UANTUM AND SEMIQUANTUM PSEUDOMETRICS 15 since R , R ∈ D ( H ) while f ∈ P ac ( R d × R d ) . By Corollary 2.4trace H ⊗ H ( P C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) → trace H ⊗ H ( P / C ̵ h ( x, z, ̵ hD x , ̵ hD z ) P / ) as ǫ → + , so that M K ̵ h ( R , R ) ≤ trace H ⊗ H ( P / C ̵ h ( x, z, ̵ hD x , ̵ hD z ) P / ) ≤ ( + α ) ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; x, ̵ hD x ) Q ( y, η ) / ) dydη + ( + α ) ∫ trace H ( Q ( y, η ) / c ̵ h ( y, η ; z, ̵ hD z ) Q ( y, η ) / ) dydη . Minimizing the right hand side of this inequality in Q ∈ C ( f, R ) and in Q ∈ C ( f, R ) shows that M K ̵ h ( R , R ) ≤ ( + α ) E ̵ h ( f, R ) + ( + α ) E ̵ h ( f, R ) . Minimizing the right hand side of this inequality over α >
0, i.e. taking α = E ̵ h ( f, R )/ E ̵ h ( f, R ) (we recall that E ̵ h ( f, R ) ≥ √ d ̵ h > M K ̵ h ( R , R ) ≤ E ̵ h ( f, R ) + E ̵ h ( f, R ) + E ̵ h ( f, R ) E ̵ h ( f, R ) = ( E ̵ h ( f, R ) + E ̵ h ( f, R )) , which is inequality (b). (cid:3) The proofs of inequalities (c)-(e) are simpler because of the rank-one assumptionon the intermediate point R . Proof of inequality (c).
According to Lemma 3.1 (ii)
M K ̵ h ( R , R ) = trace H ⊗ H (( R ⊗ R ) / C ( x, y, ̵ hD x , ̵ hD y )( R ⊗ R ) / ) M K ̵ h ( R , R ) = trace H ⊗ H (( R ⊗ R ) / C ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ) / ) since R is a rank-one density. Applying the third inequality in Lemma 4.4 showsthat C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z ) ≤ C ̵ h ( x, y, ̵ hD x , ̵ hD y ) ≤ ( + α ) C ̵ h ( x, y, ̵ hD x , ̵ hD y ) + ( + α ) C ̵ h ( y, z, ̵ hD y , ̵ hD z ) so that ( R ⊗ R ⊗ R ) / C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )( R ⊗ R ⊗ R ) / ≤ ( + α )( R ⊗ R ⊗ R ) / C ̵ h ( x, y, ̵ hD x , ̵ hD y )( R ⊗ R ⊗ R ) / + ( + α )( R ⊗ R ⊗ R ) / C ̵ h ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ⊗ R ) / = ( + α ) (( R ⊗ R ) / C ̵ h ( x, y, ̵ hD x , ̵ hD y )( R ⊗ R ) / ) ⊗ R + ( + α ) R ⊗ (( R ⊗ R ) / C ̵ h ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ) / ) . Taking the trace of both sides of this inequality in H ⊗ H ⊗ H trace H ⊗ H (( R ⊗ R ) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) = trace H ⊗ H ⊗ H (( R ⊗ R ⊗ R ) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) = trace H ⊗ H ⊗ H (( R ⊗ R ⊗ R ) / C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )( R ⊗ R ⊗ R ) / ) ≤ ( + α ) trace H ⊗ H (( R ⊗ R ) / C ̵ h ( x, y, ̵ hD x , ̵ hD y )( R ⊗ R ) / ) + ( + α ) trace H ⊗ H (( R ⊗ R ) / C ̵ h ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ) / ) = ( + α ) M K ̵ h ( R , R ) + ( + α ) M K ̵ h ( R , R ) . Passing to the limit as ǫ → + in the left hand side implies that M K ̵ h ( R , R ) ≤ trace H ⊗ H (( R ⊗ R ) / C ̵ h ( x, z, ̵ hD x , ̵ hD z )( R ⊗ R ) / ) = lim ǫ → + trace H ⊗ H (( R ⊗ R ) C ǫ ̵ h ( x, z, ̵ hD x , ̵ hD z )) ≤ ( + α ) M K ̵ h ( R , R ) + ( + α ) M K ̵ h ( R , R ) , where the first inequality follows from the definition of M K ̵ h and the fact that R ⊗ R ∈ C ( R , R ) , and the equality from Corollary 2.4.Setting α ∶ = M K ̵ h ( R , R )/ M K ̵ h ( R , R ) , we arrive at M K ̵ h ( R , R ) ≤ M K ̵ h ( R , R ) + M K ̵ h ( R , R ) + M K ̵ h ( R , R ) M K ̵ h ( R , R ) = ( M K ̵ h ( R , R ) + M K ̵ h ( R , R )) which is the inequality (c). (cid:3) Proof of inequality (d).
According to Lemma 3.1 (i) E ̵ h ( f, R ) = ∫ trace H ( R / c ̵ h ( x, ξ ) R / ) f ( x, ξ ) dxdξ E ̵ h ( g, R ) = ∫ trace H ( R / c ̵ h ( z, ζ ) R / ) g ( z, ζ ) dzdζ since R is a rank-one density. Applying the fourth inequality in Lemma 4.4 showsthat ∣ x − z ∣ + ∣ ξ − ζ ∣ ≤ ( + α ) c ̵ h ( x, ξ ; y, ̵ hD y ) + ( + α ) c ̵ h ( z, ζ ; y, ̵ hD y ) , so that (∣ x − z ∣ + ∣ ξ − ζ ∣ ) R ≤ ( + α ) R / c ̵ h ( x, ξ ) R / + ( + α ) R / c ̵ h ( z, ζ ) R / for all x, z, ξ, ζ ∈ R d . Taking the trace of both sides of this inequality, and integratingin x, ξ, z, ζ after multiplying by f ( x, ξ ) g ( z, ζ ) shows thatdist MK , ( f, g ) ≤ ∫ (∣ x − z ∣ + ∣ ξ − ζ ∣ ) f ( x, ξ ) g ( z, ζ ) dxdξdzdζ = ( + α ) ∫ trace H ( R / c ̵ h ( x, ξ ) R / ) f ( x, ξ ) dxdξ + ( + α ) ∫ trace H ( R / c ̵ h ( z, ζ ) R / ) g ( z, ζ ) dzdζ = ( + α ) E ̵ h ( f, R ) + ( + α ) E ̵ h ( g, R ) , since trace H ( R ) = ∫ f ( x, ξ ) dxdξ = ∫ g ( z, ζ ) dzdζ = . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 17
The first inequality comes from the definition of the Monge-Kantorovich-Wassersteindistance dist MK , and the fact that f ⊗ g is a (nonoptimal) coupling of f and g .Choosing α = E ̵ h ( g, R )/ E ̵ h ( f, R ) shows thatdist MK , ( f, g ) ≤ E ̵ h ( f, R ) + E ̵ h ( g, R ) + E ̵ h ( f, R ) E ̵ h ( g, R ) = ( E ̵ h ( f, R ) + E ̵ h ( g, R )) which is the inequality (d). (cid:3) Proof of inequality (e).
According to Lemma 3.1 E ̵ h ( f, R ) = ∫ trace H ( R / c ̵ h ( x, ξ ; y, ̵ hD y ) R / ) f ( x, ξ ) dxdξM K ̵ h ( R , R ) = trace H ⊗ H (( R ⊗ R ) / C ̵ h ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ) / ) since R is a rank-one density. Applying the fifth inequality in Lemma 4.4 showsthat c ̵ h ( x, ξ ; z, ̵ hD z ) ≤ ( + α ) c ̵ h ( x, ξ ; y, ̵ hD y ) + ( + α ) C ̵ h ( y, z, ̵ hD y , ̵ hD z ) so that, for each ǫ > ≤ c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) ≤ ( + α ) c ̵ h ( x, ξ ; y, ̵ hD y ) + ( + α ) C ̵ h ( y, z, ̵ hD y , ̵ hD z ) with c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) ∶ = ( I + ǫc ̵ h ( x, ξ ; z, ̵ hD z )) − c ̵ h ( x, ξ ; z, ̵ hD z ) = c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) ∗ ∈ L ( H ) for all x, ξ ∈ R d . Hence R ⊗ ( R / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) R / ) ≤ ( + α )( R / c ̵ h ( x, ξ ; y, ̵ hD y ) R / ) ⊗ R + ( + α )( R ⊗ R ) / C ̵ h ( y, z, ̵ hD y , ̵ hD z )( R ⊗ R ) / for all x, ξ ∈ R d and, taking the trace of both sides of this inequality leads to(12) trace H ( R / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) R / ) ≤ ( + α ) trace H ( R / c ̵ h ( x, ξ ; y, ̵ hD y ) R / ) + ( + α ) M K ̵ h ( R , R ) . Multiplying both sides of this inequality by f ( x, ξ ) and integrating in ( x, ξ ) showsthat ∫ trace H ( R / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) R / ) f ( x, ξ ) dxdξ ≤ ( + α ) E ̵ h ( f, R ) + ( + α ) M K ̵ h ( R , R ) = ( E ̵ h ( f, R ) + M K ̵ h ( R , R )) , with the choice α ∶ = M K ̵ h ( R , R )/ E ̵ h ( f, R ) . Since the right-hand side of (12) is integrable with respect to f ( x, ξ ) dxdξ , andtherefore finite for f ( x, ξ ) dxdξ -a.e. ( x, ξ ) ∈ R d × R d , one hastrace H ( R / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) R / ) → trace H ( R / c ̵ h ( x, ξ ; z, ̵ hD z ) R / ) for f ( x, ξ ) dxdξ -a.e. ( x, ξ ) ∈ R d × R d by Corollary 2.4. By Fatou’s lemma, observingthat f ⊗ R ∈ C ( f, R ) , one has E ̵ h ( f, R ) ≤ ∫ trace H ( R / c ̵ h ( x, ξ ; z, ̵ hD z ) R / ) f ( x, ξ ) dxdξ ≤ lim ǫ → + ∫ trace H ( R / c ǫ ̵ h ( x, ξ ; z, ̵ hD z ) R / ) f ( x, ξ ) dxdξ ≤ ( E ̵ h ( f, R ) + M K ̵ h ( R , R )) , which is the inequality (e). (cid:3) Applications
One satisfying consequence of the triangle inequalities proved in the last sectionis the following statement, which confirms that
M K ̵ h can indeed be thought of asa quantum deformation of the quadratic Monge-Kantorovich-Wasserstein distance. Theorem 5.1.
Let R ̵ h , S ̵ h be families of density operators in D ( H ) , and let f, g ∈ P ac ( R d × R d ) . Assume that E ̵ h ( f, R ̵ h ) → and E ̵ h ( g, S ̵ h ) → as ̵ h → . Then lim ̵ h → M K ̵ h ( R ̵ h , S ̵ h ) = dist MK , ( f, g ) . This statement is to be compared with the lower bound
M K ̵ h ( R ̵ h , S ̵ h ) ≥ dist MK , (̃ W ̵ h ( R ̵ h ) , ̃ W ̵ h ( S ̵ h )) − d ̵ h , which is Theorem 2.3 (2) in [6], and with the upper bound obtained in the specialcase of T¨oplitz operators M K ̵ h ( OP T ̵ h (( π ̵ h ) d µ ) , OP T ̵ h (( π ̵ h ) d ν )) ≤ dist MK , ( µ, ν ) + d ̵ h , stated as Theorem 2.3 (1) in [6]. Proof.
By Theorem 4.2 (a)-(b),
M K ̵ h ( R ̵ h , S ̵ h ) ≤ E ̵ h ( f, R ̵ h ) + E ̵ h ( f, S ̵ h ) ≤ E ̵ h ( f, R ̵ h ) + dist MK , ( f, g ) + E ̵ h ( g, S ̵ h ) . Hence lim ̵ h → + M K ̵ h ( R ̵ h , S ̵ h ) ≤ dist MK , ( f, g ) . By Theorem 2.4 (2) in [7]dist MK , ( f, ̃ W ( R ̵ h )) ≤ E ̵ h ( f, R ̵ h ) + d ̵ h (notice the slight change of normalization in the definition of E ̵ h between [7] andthe present paper), so that our assumption implies thatdist MK , ( f, ̃ W ( R ̵ h )) → MK , ( g, ̃ W ( S ̵ h )) → ̵ h →
0. From the inequalitydist MK , (̃ W ̵ h ( R ̵ h ) , ̃ W ̵ h ( S ̵ h )) ≤ M K ̵ h ( R ̵ h , S ̵ h ) + d ̵ h , (Theorem 2.3 (2) in [6]), we deduce thatdist MK , ( f, g ) ≤ lim ̵ h → M K ̵ h ( R ̵ h , S ̵ h ) . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 19
Notice that this last lower bound is a variant of the last inequality in Theorem2.3 of [6], except that in the present case the assumption on R ̵ h and S ̵ h is slightlydifferent (in other words, we have assumed that E ̵ h ( f, R ̵ h ) → ̃ W ̵ h ( R ̵ h ) → f in S ′ ( R d × R d ) .) (cid:3) Kantorovich duality for E ̵ h Theorem 6.1.
Let S ∈ D ( H ) and let p ≡ p ( x, ξ ) be a probability density on R d such that ∫ R d (∣ x ∣ + ∣ ξ ∣ ) p ( x, ξ ) dxdξ < +∞ . Then E ̵ h ( p, S ) = min Q ∈ C( p,S ) ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ = sup a ∈ Cb ( R d ) ,B = B ∗∈L( H ) a ( x,ξ ) I H + B ≤ c ( x,ξ ) ( ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( BS )) . Notice that the duality theorem implies in particular the existence of at leastone optimal coupling Q ∈ C ( p, S ) . Proof.
The proof is split in several steps.
Step 1: the functions f and g . Consider the Banach space E ∶ = C b ( R d ; L ( H )) ,with ∥ T ∥ E ∶ = sup ( x,ξ ) ∈ R d ∥ T ( x, ξ )∥ , and set f ( T ) ∶ = ⎧⎪⎪⎨⎪⎪⎩ T ( x, ξ ) = T ( x, ξ ) ∗ ≥ − c ( x, ξ ) for all ( x, ξ ) ∈ R d , + ∞ otherwise,while g ( T ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ∫ R d ap ( x, ξ ) dxdξ + trace H ( BS ) if T ( x, ξ ) = T ( x, ξ ) ∗ = a ( x, ξ ) I H + B for all ( x, ξ ) ∈ R d , + ∞ otherwise,The constraint T ( x, ξ ) = T ( x, ξ ) ∗ ≥ − c ( x, ξ ) means that, for each ( x, ξ ) ∈ R d , onehas ⟨ φ ( x, ξ )∣ T ( x, ξ ) + c ( x, ξ )∣ φ ( x, ξ )⟩ ≥ φ ∈ Form-Dom ( c ( x, ξ )) . On the other hand, the nullspace of the linear map C b ( R d ) × L ( H ) ∋ ( a, B ) ↦ Γ ( a, B ) ≡ a ( x, ξ ) I H + B ∈ E is Ker ( L ) = {( t, − tI H ) , t ∈ R } . Since g (( a + t ) I H + ( B − tI H )) = g ( aI H + B ) + t ∫ R d p ( x, ξ ) dxdξ − t trace H ( S ) = g ( aI H + B ) , the prescription above defines g on Ran ( Γ ) ≃ ( C b ( R d ) × L ( H ))/ Ker ( Γ ) . Observethat g (( aI H + B ) ∗ ) = g ( ¯ aI H + B ∗ ) = ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( B ∗ S ) =∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H (( SB ) ∗ ) =∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( SB ) = g ( aI H + B ) , so that ( aI H + B ) ∗ = aI H + B Ô⇒ g ( aI H + B ) ∈ R . Thus the definition aboveimplies that g takes its values in ( −∞ , +∞ ] .The functions f and g are convex. Indeed, g is the extension by +∞ of a R -linearfunctional defined on the set of self-adjoint elements of Ran ( Γ ) , which is a linearsubspace of E . As for f , it is the indicator function (in the sense of the definitionin § { T ∈ E s.t. T ( x, ξ ) = T ( x, ξ ) ∗ ≥ − c ( x, ξ ) for all ( x, ξ ) ∈ R d } and is therefore convex. Besides f ( ) = g ( ) =
0, and f is continuous at 0. Indeed,by the Heisenberg inequality c ( x, ξ ) ≥ d ̵ hI H for all ( x, ξ ) ∈ R d , so that, for each T ∈ ET ( x, ξ ) = T ( x, ξ ) ∗ and ∥ T ( x, ξ )∥ < d ̵ h for all ( x, ξ ) ∈ R d Ô⇒ T ( x, ξ ) ≥ − c ( x, ξ ) for all ( x, ξ ) ∈ R d Ô⇒ f ( T ) = . In particular f is continuous at 0. Step 2: applying convex duality.
By the Fenchel-Rockafellar convex duality theorem(Theorem 1.12 in [1])inf T ∈ E ( f ( T ) + g ( T )) = max Λ ∈ E ′ ( − f ∗ ( − Λ ) − g ∗ ( Λ )) . Let us compute the Legendre duals f ∗ and g ∗ .First f ∗ ( − Λ ) = sup T ∈ E (⟨ − Λ , T ⟩ − f ( T )) = sup T ∈ ET ( x,ξ )= T ( x,ξ )∗≥− c ( x,ξ ) ⟨ − Λ , T ⟩ . If Λ ∈ E ′ is not a nonnegative linear functional, there exists T ∈ E such that T ( x, ξ ) = T ( x, ξ ) ∗ ≥ ⟨ Λ , T ⟩ = − α <
0. Since nT ( x, ξ ) = nT ( x, ξ ) ∗ ≥ ≥ − d ̵ hI H ≥ − c ( x, ξ ) for all ( x, ξ ) ∈ R d one has f ∗ ( − Λ ) ≥ sup n ≥ ⟨ − Λ , nT ⟩ = sup n ≥ ( nα ) = +∞ . For Λ ∈ E ′ such that Λ ≥
0, we define ⟨ Λ , c ⟩ ∶ = sup T ∈ ET ( x,ξ )= T ( x,ξ )∗≤ c ( x,ξ ) ⟨ Λ , T ⟩ ∈ [ , +∞ ] . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 21 (Observe indeed that T = c ( x, ξ ) = c ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d .) With this definition, one has clearly f ∗ ( − Λ ) ∶ = {⟨ Λ , c ⟩ if Λ ≥ , g ∗ ( Λ ) = sup T ∈ E (⟨ Λ , T ⟩ − g ( T )) = sup T ∈ ET ( x,ξ )= T ( x,ξ )∗= a ( x,ξ ) I H + B (⟨ Λ , T ⟩ − ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ − trace H ( BS )) . If there exists a ≡ a ( x, ξ ) ∈ C b ( R d , R ) and B = B ∗ ∈ L ( H ) such that either ⟨ Λ , aI H + B ⟩ > ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( BS ) or ⟨ Λ , aI H + B ⟩ < ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( BS ) , one has either g ( Λ ) ≥ sup n ≥ (⟨ Λ , n ( aI H + B )⟩ − n ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ − n trace H ( BS )) = +∞ , or g ( Λ ) ≥ sup n ≥ (⟨ Λ , n ( − aI H − B )⟩ + n ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + n trace H ( BS )) = +∞ . Hence g ∗ ( Λ ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ⟨ Λ , aI H + B ⟩ = ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( BS ) for each a ≡ a ( x, ξ ) ∈ C b ( R d , R ) and B = B ∗ ∈ L ( H ) , + ∞ otherwise.Notice that the prescription ⟨ Λ , aI H + B ⟩ = ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( BS ) defines a unique linear functional on the set of T ∈ Ran Γ such that T ( x, ξ ) ∗ = T ( x, ξ ) for each ( x, ξ ) ∈ R d by the same argument as in Step 1.Therefore, the Fenchel-Rockafellar duality theorem in this case results in theequalityinf T ∈ E ( f ( T ) + g ( T )) = inf a ∈ Cb ( R d, R ) ,B = B ∗ a ( x,ξ ) I H + B ≥− c ( x,ξ ) ( ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( SB )) = max Λ ∈ E ′ ( f ∗ ( − Λ ) + g ∗ ( Λ )) = max ≤ Λ ∈ E ′ , ⟨ Λ ,aI H + B ⟩=∫ a ( x,ξ ) p ( x,ξ ) dxdξ + trace ( SB ) − ⟨ Λ , c ⟩ or, equivalently sup a ∈ Cb ( R d, R ) ,B = B ∗ a ( x,ξ ) I H + B ≤ c ( x,ξ ) ( ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( SB )) = min ≤ Λ ∈ E ′ , ⟨ Λ ,aI H + B ⟩=∫ a ( x,ξ ) p ( x,ξ ) dxdξ + trace ( SB ) ⟨ Λ , c ⟩ . Step 3: representing the optimal Λ . Define a linear map F Λ ∶ C b ( R d ) → L ( H ) bythe formula trace H ( KF Λ ( a )) = Λ ( aK ) , for each K ∈ K ( H ) . Indeed, since K ↦ Λ ( aK ) is a linear functional on K ( H ) which is continuous forthe norm topology, and since K ( H ) ′ = L ( H ) , this linear functional is representedby a trace-class operator F Λ ( a ) . Since Λ is linear, the map F Λ is linear.Since Λ ≥
0, one has F Λ ( a ) = F Λ ( a ) ∗ ≥ a ∈ C b ( R d ) such that a ( x, ξ ) ≥ ( x, ξ ) ∈ R d . Indeed, for a ∈ C b ( R d ; R ) , set T ∶ = ( F Λ ( a ) + F Λ ( a ) ∗ ) , T ∶ = − i ( F Λ ( a ) − F Λ ( a ) ∗ ) . Then, for each K = K ∗ ∈ K ( H ) , one hasΛ ( aK ) = trace H ( T K ) + i trace H ( T K ) with T r H ( T j K ) = T r H (( T j K ) ∗ ) = T r H ( K ∗ T ∗ j ) = T r H ( KT j ) = T r H ( T j K ) for j = ,
2. Since a ∈ C b ( R d ; R ) and K = K ∗ ∈ L ( H ) , one has − ∥ a ∥ L ∞ ∥ K ∥ I H ≤ aK ≤ ∥ a ∥ L ∞ ∥ K ∥ I H so that − ∥ a ∥ L ∞ ∥ K ∥ ≤ Λ ( aK ) ≤ ∥ a ∥ L ∞ ∥ K ∥ since Λ ( I H ) = ∫ R d p ( x, ξ ) dxdξ = . In particular, Λ ( aK ) ∈ R , so that trace H ( T K ) = K = K ∗ ∈ K ( H ) . Since T = T ∗ ∈ L ( H ) , specializing this identity to the case where K is the orthogonalprojection on any eigenvector of T shows that T =
0. Thus a ∈ C b ( R d ; R ) Ô⇒ F Λ ( a ) = F Λ ( a ) ∗ . Moreover a ∈ C b ( R d ; R ) and a ≥ Ô⇒ trace H ( F Λ ( a ) K ) ≥ K = K ∗ ≥ K ( H ) and specializing this last inequality to the case where K is the orthogonal projectionon any eigenvector of F Λ ( a ) = F Λ ( a ) ∗ ∈ L ( H ) shows that all the eigenvalues of F Λ ( a ) are nonnegative, so that F Λ ( a ) ≥ F Λ , i.e.Λ ( aK ) = trace H ( F Λ ( a ) K ) for each a ∈ C b ( R d ; C ) and K ∈ K ( H ) that ∥ F Λ ( a )∥ ≤ ∥ Λ ∥∥ a ∥ L ∞ ( R d ) . Next we specialize this defining identity to the case where a ≥ R d while K = Π n is the orthogonal projection on span { e , . . . , e n } , with ( e , e , . . . ) a completeorthonormal system in H . One hasΛ ( a Π n ) = trace H ( F Λ ( a ) Π n ) → trace H ( F Λ ( a )) = ∥ F Λ ( a )∥ as n → ∞ while a ( I H − Π n ) ≥ ( a Π n ) ≤ Λ ( aI H ) = ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ so that a ∈ C b ( R d ) and a ≥ Ô⇒ ∥ F Λ ( a )∥ ≤ ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ . UANTUM AND SEMIQUANTUM PSEUDOMETRICS 23
More generally, for each a ∈ C b ( R d ; R ) , one has − ∣ a ∣ ≤ a ≤ ∣ a ∣ so that ∣ trace H ( F Λ ( a )∣ e j ⟩⟨ e j ∣)∣ = ∣ Λ ( a ∣ e j ⟩⟨ e j ∣)∣ ≤ Λ (∣ a ∣∣ e j ⟩⟨ e j ∣) for each j ≥
1, where ( e , e , . . . , ) is a complete orthonormal system of eigenvectorsof F Λ ( a ) = F Λ ( a ) ∗ ∈ L ( H ) . Hence n ∑ j = ∣ trace H ( F Λ ( a )∣ e j ⟩⟨ e j ∣)∣ ≤ Λ ⎛⎝∣ a ∣ n ∑ j = ∣ e j ⟩⟨ e j ∣⎞⎠ ≤ Λ (∣ a ∣ I H ) , and since n ∑ j = ∣ trace H ( F Λ ( a )∣ e j ⟩⟨ e j ∣)∣ → ∥ F Λ ( a )∥ as n → ∞ we conclude that ∥ F Λ ( a )∥ ≤ Λ (∣ a ∣ I H ) = ∫ R d ∣ a ( x, ξ )∣ p ( x, ξ ) dxdξ . Since C b ( R d ) is dense in L ( R d , pdxdξ ) , this inequality, applied to the real andthe imaginary part of a , shows that F Λ is a continuous linear operator from L ( R d to L ( H ) . Since L ( H ) is separable and is the dual of the Banach space K ( H ) (thenorm closure in L ( H ) of the set of finite rank operators), we conclude from theDunford-Pettis theorem (Theorem 1 in § L ( H ) has theRadon-Nikodym property. By Theorem 5 in § F Λ is Riesz-representable: in other words, there exists q ∈ L ∞ ( R d , pdxdξ ; L ( H )) such that F Λ ( a ) = ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ , for all a ∈ L ( R d , pdxdξ ) . Step 4: defining the optimal coupling.
We have seen that a ∈ C b ( R d ) and a ≥ Ô⇒ F Λ ( a ) = ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ ≥ . This implies that q ( x, ξ ) = q ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d .Next, one hasΛ ( K ) = trace H ( F Λ ( ) K ) = trace H ( KS ) , K ∈ K ( H ) , so that F Λ ( ) = ∫ R d q ( x, ξ ) p ( x, ξ ) dxdξ = S ∈ L ( H ) = K ( H ) ′ . On the other hand, for each a ∈ C b ( R d ) such that a ≥
0, one hastrace H ( P n ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ ) = trace H ( F Λ ( a ) P n ) = Λ ( aP n ) ≤ Λ ( aI H ) = ∫ R d a ( x, ξ ) d ( x, ξ ) dxdξ where P n is the orthogonal projection on span { e , . . . , e n } , with ( e , e , . . . ) beinga complete orthonormal system of eigenvectors of ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ ∈ L ( H ) . Letting n → ∞ , one hastrace H ( P n ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ ) → trace H ( ∫ R d a ( x, ξ ) q ( x, ξ ) p ( x, ξ ) dxdξ ) , so that ∫ R d a ( x, ξ ) trace H ( q ( x, ξ )) p ( x, ξ ) dxdξ ≤ ∫ R d a ( x, ξ ) d ( x, ξ ) dxdξ . Since this holds for each a ∈ C b ( R d such that a ≥
0, we conclude thattrace H ( q ( x, ξ )) ≤ p ( x, ξ ) dxdξ –a.e. ( x, ξ ) ∈ R d . Moreover ∫ R d ( − trace H ( q ( x, ξ ))) p ( x, ξ ) dxdξ = − trace H ( S ) = H ( q ( x, ξ )) = p ( x, ξ ) dxdξ –a.e. ( x, ξ ) ∈ R d . In other words, we have proved that ( x, ξ ) ↦ Q ( x, ξ ) = p ( x, ξ ) q ( x, ξ ) defines anelement of C ( p, S ) . Step 5: extending the representation formula for Λ . For each B ∈ E , we define ⟨ L, B ⟩ ∶ = ⟨ Λ , B ⟩ − ∫ R d trace H ( B ( x, ξ ) Q ( x, ξ )) dxdξ . Let us prove that B ∈ E and B ( x, ξ ) = B ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d Ô⇒ ⟨ L, B ⟩ ≥ . Pick ǫ >
0, and let Q ǫ be a simple L ( H ) -valued function on R d such that ∫ R d ∥ Q ( x, ξ ) − Q ǫ ( x, ξ )∥ dxdξ < ǫ . Write Q ǫ ( x, ξ ) = N ∑ j = Ω j ( x, ξ ) Q j , ≤ Q j = Q ∗ j ∈ L ( H ) for each j = , . . . , N , where Ω j are bounded, pairwise disjoint measurable sets in R d for j = , . . . , N .For each j = , . . . , N , let ( e j, , e j, , . . . ) designate a complete orthonormal system ofeigenvectors of Q j , and let P j,n be the orthogonal projection on span { e j, , . . . , e j,n } .Define Π n ( x, ξ ) = N ∑ j = Ω j ( x, ξ ) P j,n . One easily checks that Π n ( x, ξ ) = Π n ( x, ξ ) ∗ = Π n ( x, ξ ) for each ( x, ξ ) ∈ R d . Then,for each B ∈ E such that B ( x, ξ ) = B ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d , one has0 ≤ ⟨ Λ , ( I H − Π n ) B ( I H − Π n )⟩ = ⟨ Λ , B ⟩ − ⟨ Λ , Π n B + B Π n − Π n B Π n ⟩ = ⟨ Λ , B ⟩ − ∫ R d trace H (( Π n B + B Π n − Π n B Π n ) Q )( x, ξ ) dxdξ = ⟨ Λ , B ⟩ − ∫ R d trace H (( Π n B + B Π n − Π n B Π n ) Q ǫ )( x, ξ ) dxdξ + ∫ R d trace H (( Π n B + B Π n − Π n B Π n )( Q ǫ − Q ))( x, ξ ) dxdξ . By construction, keeping ǫ > ∫ R d trace H (( Π n B + B Π n − Π n B Π n ) Q ǫ )( x, ξ ) dxdξ = ∫ R d trace H ( B Π n Q ǫ Π n )( x, ξ ) dxdξ → ∫ R d trace H ( BQ ǫ )( x, ξ ) dxdξ UANTUM AND SEMIQUANTUM PSEUDOMETRICS 25 as n → ∞ , so that0 ≤ lim n → ∞ ⟨ Λ , ( I H − Π n ) B ( I H − Π n )⟩ = ⟨ Λ , B ⟩ − ∫ R d trace H ( BQ )( x, ξ ) dxdξ + lim n → ∞ ∫ R d trace H (( Π n B + B Π n − Π n B Π n )( Q ǫ − Q ))( x, ξ ) dxdξ + ∫ R d trace H ( B ( Q − Q ǫ ))( x, ξ ) dxdξ . On the other hand ∣ ∫ R d trace H (( Π n B + B Π n − Π n B Π n )( Q ǫ − Q ))( x, ξ ) dxdξ ∣ ≤ ∫ R d ∣ trace H (( Π n B + B Π n − Π n B Π n )( Q ǫ − Q ))( x, ξ )∣ dxdξ ≤ ∫ R d ∥( Π n B + B Π n − Π n B Π n )( x, ξ )∥∥( Q ǫ − Q )( x, ξ )∥ dxdξ ≤ ( x,ξ ) ∈ R d ∥ B ( x, ξ )∥ ∫ R d ∥( Q ǫ − Q )( x, ξ )∥ dxdξ ≤ ǫ sup ( x,ξ ) ∈ R d ∥ B ( x, ξ )∥ while, by the same token, ∣ ∫ R d trace H ( B ( Q − Q ǫ ))( x, ξ ) dxdξ ∣ ≤ ǫ sup ( x,ξ ) ∈ R d ∥ B ( x, ξ )∥ . Finally ⟨ Λ , B ⟩ − ∫ R d trace H ( BQ )( x, ξ ) dxdξ ≥ − ǫ sup ( x,ξ ) ∈ R d ∥ B ( x, ξ )∥ and since this holds for each ǫ >
0, we conclude that B ∈ E and B ( x, ξ ) = B ( x, ξ ) ∗ ≥ ( x, ξ ) ∈ R d Ô⇒ ⟨ L, B ⟩ ≥ . By a classical argument, this implies that ∥ L ∥ = ⟨ L, I H ⟩ .On the other hand ⟨ L, I H ⟩ = ⟨ L, I H ⟩ − ∫ R d trace H ( q ( x, ξ )) p ( x, ξ ) dxdξ = trace H ( S ) − ∫ R d p ( x, ξ ) dxdξ = L =
0. In other words, the representation formula ⟨ Λ , B ⟩ = ∫ R d trace H ( B ( x, ξ ) Q ( x, ξ )) dxdξ holds for each B ∈ E , and not only for B ∈ C b ( R d ; K ( H )) . Step 6: computing ⟨ Λ , c ⟩ . As explained in Step 2 ⟨ Λ , c ⟩ = sup T ∈ ET ( x,ξ )= T ( x,ξ )∗≤ c ( x,ξ ) ⟨ Λ , T ⟩ . For each n ≥
1, set c n ( x, ξ ) ∶ = ( I H + n c ( x, ξ )) − c ( x, ξ ) ∈ L ( H ) , so that0 ≤ c ( x, ξ ) = c ( x, ξ ) ∗ ≤ . . . ≤ c n ( x, ξ ) = c n ( x, ξ ) ∗ ≤ . . . ≤ c ( x, ξ ) = c ( x, ξ ) ∗ . Thus, by definition ⟨ Λ , c n ⟩ = ∫ R d trace H ( Q ( x, ξ ) c n ( x, ξ )) dxdξ ≤ ⟨ Λ , c ⟩ for each n ≥
1, so that, by Corollary 2.4 ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ = lim n → ∞ ∫ R d trace H ( Q ( x, ξ ) c n ( x, ξ )) dxdξ ≤ ⟨ Λ , c ⟩ . On the other hand, let ( e ( x, ξ ) , e ( x, ξ ) , . . . , ) designate a complete orthonormalsystem in H of eigenfunctions of c ( x, ξ ) , with c ( x, ξ ) e j ( x, ξ ) = λ j e j ( x, ξ ) for j ≥ c ( x, ξ ) is a phase space translate of the harmonic oscillator H ∶ = (∣ x ∣ − ̵ h ∆ x ) ,the eigenvalues λ j are independent of ( x, ξ ) . Set t kl ( x, ξ ) ∶ = ⟨ e k ( x, ξ )∣ Q ( x, ξ ) / ∣ e l ( x, ξ )⟩ , k, l ≥ . Since ( x, ξ ) ↦ Q ( x, ξ ) / ∈ L ( R d ; L ( H )) , one has v k ( x, ξ ) ∶ = ∑ l ≥ t kl ( x, ξ ) e l ( x, ξ ) ∈ Form-Dom ( c ( x, ξ )) for a.e. ( x, ξ ) ∈ R d and trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) = ∑ k,l ≥ λ l ∣ t kl ( x, ξ )∣ = ∑ k ≥ ⟨ v k ( x, ξ )∣ c ( x, ξ )∣ v k ( x, ξ )⟩ < ∞ for a.e. ( x, ξ ) ∈ R d , since ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ < ∞ . Taking this last inequality for granted, we conclude as follows. Let a ≡ a ( x, ξ ) ∈ C b ( R d ) and B = B ∗ ∈ L ( H ) satisfy the constraint a ( x, ξ ) I H + B ≤ c ( x, ξ ) , ( x, ξ ) ∈ R d in the sense that a ( x, ξ )∥ φ ∥ H + ⟨ φ ∣ B ∣ φ ⟩ ≤ ⟨ φ ∣ c ( x, ξ )∣ φ ⟩ for each φ ∈ Form-Dom ( c ( x, ξ )) . Since v k ( x, ξ ) ∈ Form-Dom ( c ( x, ξ )) for a.e. ( x, ξ ) ∈ R d and each k ≥ a ( x, ξ ) p ( x, ξ ) + trace H ( Q ( x, ξ ) B ) = a ( x, ξ ) trace H ( Q ( x, ξ )) + trace H ( Q ( x, ξ ) / BQ ( x, ξ ) / ) = a ( x, ξ ) ∑ k ≥ ⟨ v k ( x, ξ )∣ v k ( x, ξ )⟩ + ∑ k ≥ ⟨ v k ( x, ξ )∣ B ∣ v k ( x, ξ )⟩ ≤ ∑ k ≥ ⟨ v k ( x, ξ )∣ c ( x, ξ )∣ v k ( x, ξ )⟩ = trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) . Integrating in ( x, ξ ) shows that ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( SB ) ≤ ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ UANTUM AND SEMIQUANTUM PSEUDOMETRICS 27 since, by construction, ∫ R d Q ( x, ξ ) dxdξ = S .
Thus ⟨ Λ , c ⟩ = sup a ∈ Cb ( R d ) ,B = B ∗∈L( H ) a ( x,ξ ) I H + B ≤ c ( x,ξ ) ( ∫ R d a ( x, ξ ) p ( x, ξ ) dxdξ + trace H ( SB )) ≤ ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ ≤ ⟨ Λ , c ⟩ , where the first equality follows from convex duality as explained in Step 2, while thelast inequality has been obtained above at the beginning of Step 6. This completesthe proof.It remains to prove that ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ < ∞ . Since c ( x, ξ ) ≤ (∣ x ∣ + ∣ ξ ∣ ) I H + H one has v k ( x, ξ ) ∈ Form-Dom ( H ) Ô⇒ v k ( x, ξ ) ∈ Form-Dom ( c ( x, ξ )) and ⟨ v k ( x, ξ )∣ c ( x, ξ )∣ v k ( x, ξ )⟩ ≤ (∣ x ∣ + ∣ ξ ∣ )∥ v k ( x, ξ )∥ H + ⟨ v k ( x, ξ )∣ H ∣ v k ( x, ξ )⟩ . Let ( h , h , . . . ) be a complete orthonormal system of eigenvectors of H in H (theHermite functions), with eigenvalues µ j . Since ∑ k ≥ t km ( x, ξ ) t kn ( x, ξ ) = ⟨ e m ( x, ξ )∣ Q ( x, ξ )∣ e n ( x, ξ )⟩ by definition of t kl ( x, ξ ) , one has ∑ k ≥ ∣⟨ v k ( x, ξ )∣ h j ⟩∣ = ∑ k ≥ ∑ m,n ≥ t km ( x, ξ ) t kn ( x, ξ )⟨ e m ( x, ξ )∣ h j ⟩⟨ h j ∣ e n ( x, ξ )⟩ = ∑ m,n ≥ ⟨ e m ( x, ξ )∣ Q ( x, ξ )∣ e n ( x, ξ )⟩⟨ e m ( x, ξ )∣ h j ⟩⟨ h j ∣ e n ( x, ξ )⟩ = ⟨ h j ∣ Q ( x, ξ )∣ h j ⟩ . Hence ∫ R d ∑ k ≥ ⟨ v k ( x, ξ )∣ H ∣ v k ( x, ξ )⟩ dxdξ = ∑ j ≥ µ j ∫ R d ⟨ h j ∣ Q ( x, ξ )∣ h j ⟩ = ∑ j ≥ µ j ⟨ h j ∣ S ∣ h j ⟩ = trace H ( S / ∣ H ∣ S / ) < ∞ , and since ∑ k ≥ ∥ v k ( x, ξ )∥ H = trace H ( Q ( x, ξ )) = p ( x, ξ ) , one concludes that ∫ R d trace H ( Q ( x, ξ ) / c ( x, ξ ) Q ( x, ξ ) / ) dxdξ ≤ ∫ R d (∣ x ∣ + ∣ ξ ∣ ) p ( x, ξ ) dxdξ + trace H ( S / HS / ) < ∞ . (cid:3) Applications of duality for E ̵ h I: inequalities between
M K ̵ h , E ̵ h and dist MK , . Theorem 7.1.
Let
R, S ∈ D ( H ) and p be a probability density on R d . Then E ̵ h (̃ W ̵ h ( R ) , S ) ≥ dist MK , (̃ W ̵ h [ R ] , ̃ W ̵ h [ S ] − d ̵ h,M K ̵ h ( R, S ) ≥ E ̵ h (̃ W ̵ h ( R ) , S ) − d ̵ h ,M K ̵ h ( R, S ) ≥ dist MK , (̃ W ̵ h [ R ] , ̃ W ̵ h [ S ] − d ̵ h . Proof.
The first inequality and the third inequality (also a consequence of the twoothers) were proved in Theorem 2.4 (2) of [7] and Theorem 2.3 (2) of [6] respectively.The second inequality is proved along the same lines as Theorem 2.3 (2) of [6].Let a ≡ a ( x, ξ ) in C b ( R d ; R ) and B = B ∗ ∈ L ( H ) satisfy a ( x, ξ ) I H + B ≤ c ( x, ξ ) for a.e. ( x, ξ ) ∈ R d . Then a ( x, ξ )∣ x, ξ ⟩⟨ x, ξ ∣ ⊗ I H + ∣ x, ξ ⟩⟨ x, ξ ∣ ⊗ B ≤ ∣ x, ξ ⟩⟨ x, ξ ∣ ⊗ c ( x, ξ ) for a.e. ( x, ξ ) ∈ R d , so thatOP T ̵ h (( π ̵ h ) d a ) ⊗ I H + I H ⊗ B ≤ ( π ̵ h ) d ∫ R d ∣ x, ξ ⟩⟨ x, ξ ∣ ⊗ c ( x, ξ ) dxdξ = C + d ̵ hI H ⊗ H . Thus, for each Q ∈ C ( R, S ) , one hastrace H ⊗ H ( Q / C Q / ) + d ̵ h ≥ trace H ⊗ H ( Q / ( OP T ̵ h (( π ̵ h ) d a ) ⊗ I H + I H ⊗ B ) Q / ) = trace H ⊗ H ( Q ( OP T ̵ h (( π ̵ h ) d a ) ⊗ I H + I H ⊗ B )) = trace H ( R OP T ̵ h (( π ̵ h ) d a ) + SB ) = ∫ R d a ( x, ξ )̃ W ̵ h ( R )( x, ξ ) dxdξ + trace H ( SB ) . In particular
M K ̵ h ( R, S ) + d ̵ h = inf Q ∈ C( R,S ) trace H ⊗ H ( Q / C Q / ) + d ̵ h ≥ sup a ∈ Cb ( R d, R ) ,B = B ∗∈L( H ) a ( x,ξ ) I H + B ≤ c ( x,ξ ) ( ∫ R d a ( x, ξ )̃ W ̵ h ( R )( x, ξ ) dxdξ + trace H ( SB )) = E ̵ h (̃ W ̵ h ( R ) , S ) . (cid:3) Applications of duality for E ̵ h II: “triangle” inequalities
Theorem 8.1.
Let
R, S, T ∈ D ( H ) and let f, g ∈ P ( R d ) . Then(i) one has dist MK , ( f, g ) ≤ √ E ̵ h ( f, S ) + d ̵ h + √ E ̵ h ( g, S ) + d ̵ h < E ̵ h ( f, S ) + E ̵ h ( g, S ) + d ̵ h ; (ii) one has E ̵ h ( f, T ) ≤ dist MK , ( f, ̃ W ̵ h ( S )) + E ̵ h (̃ W ̵ h ( S ) , T ) ≤ √ E ̵ h ( f, S ) + d ̵ h + √ M K ̵ h ( S, T ) + d ̵ h < E ̵ h ( f, S ) + M K ̵ h ( S, T ) + d ̵ h ; UANTUM AND SEMIQUANTUM PSEUDOMETRICS 29 (iii) one has
M K ̵ h ( R, T ) ≤ E ̵ h (̃ W ̵ h ( S ) , R ) + E ̵ h (̃ W ̵ h ( S ) , T ) ≤ √ M K ̵ h ( R, S ) + d ̵ h + √ M K ̵ h ( S, T ) + d ̵ h < M K ̵ h ( R, S ) + M K ̵ h ( S, T ) + d ̵ h . Proof.
The triangle inequality for dist MK , implies thatdist MK , ( f, g ) ≤ dist MK , ( f, ̃ W ̵ h ( S )) + dist MK , (̃ W ̵ h ( S ) , g ) . Then, Theorem 2.4 (2) of [7] implies thatdist MK , ( f, ̃ W ̵ h ( S )) ≤ √ E ̵ h ( f, S ) + d ̵ h , dist MK , (̃ W ̵ h ( S ) , g ) ≤ √ E ̵ h ( g, S ) + d ̵ h . This implies the first inequality in (i). As for the second inequality, for each
X, Y >
0, one has the obvious elementary inequality √ X + Y < X + Y .
This inequality obviously applies to the present case since E ̵ h ( f, S ) ≥ d ̵ h and E ̵ h ( g, S ) ≥ d ̵ h by Theorem 2.4 (2) of [7]. This proves (i).Observe that the first inequality in (ii) is inequality (a) in Theorem 4.2 with g = ̃ W ̵ h ( S ) and R = T . Then Theorem 2.4 (2) of [7] implies thatdist MK , ( f, ̃ W ̵ h ( S )) ≤ √ E ̵ h ( f, S ) + d ̵ h , while Theorem 7.1 implies that E ̵ h (̃ W ̵ h ( S ) , T ) ≤ √ M K ̵ h ( S, T ) + d ̵ h , and this implies the second inequality in (ii). The third inequality is obtained asin (i).Finally, the first inequality in (iii) is inequality (b) in Theorem 4.2 with R = R ,while R = T and f = ̃ W ̵ h ( S ) . Then, Theorem 7.1 implies that E ̵ h (̃ W ̵ h ( S ) , R ) ≤ √ M K ̵ h ( R, ST ) + d ̵ h , E ̵ h (̃ W ̵ h ( S ) , T ) ≤ √ M K ̵ h ( S, T ) + d ̵ h , which gives the second inequality in (iii). Finally, the third inequality is obtainedas in (i). (cid:3) Remark.
It is interesting to compare the inequality (iii) above with the “genere-liazed triangle inequality” in [4]. Let us recall that DePalma and Trevisan haveconstructed a pseudo-distance on density operators on H which is similar to oursto some extent. The DePalma-Trevisan distance D is defined through a differentnotion of coupling than in [6]; specifically, their notion of couplings is based on“quantum channels” (completely positive linear maps on the set of density oper-ators): see Definition 1 in [4]. While the transport cost in formula (19) of [4] isin some sense reminiscent of the transport cost used in [6], these two costs are infact significantly different. For instance, the transport cost used in the definitionof M K ̵ h in [6], and in the present paper, has compact resolvent, and therefore itsspectrum consists of eigenvalues only. On the contrary, the cost operator in [4] inthe case of Gaussian quantum systems has continuous spectrum on [ , +∞ ) . In Theorem 2 of [4], DePalma and Trevisan prove what they call a “triangleinequality” for their distance D , of the form D ( R, T ) ≤ D ( R, S ) + D ( S, S ) + D ( S, T ) (inequality (35) in [4]). Of course, if D was a real distance, D ( S, S ) =
0, and theinequality above coincides with the usual triangle inequality. In [4], there is anexplicit formula for D ( S, S ) in terms of the canonical purification of S (Corollary1, formula (34) in [4]).With the distance M K ̵ h defined in [6], one has M K ̵ h ( R, S ) ≥ d ̵ h , for all R, S ∈ D ( H ) , so that Theorem 8.1 (iii) implies that M K ̵ h ( R, T ) < M K ̵ h ( R, S ) + M K ̵ h ( S, S ) + M K ̵ h ( S, T ) . In other words,
M K ̵ h satisfies the same “generalized triangle inequality” as theDePalma-Trevisan distance D , with a strict inequality.9. Applications of duality for E ̵ h III: Classical/quantum optimaltransport and semiquantum Legendre transform
A classical/quantum optimal transport.
Let r be a probability densityon R d and S a density operator on L ( R d ) .We suppose that an optimal operator ̃ B and an optimal function ̃ a exists forthe Kantorovich duality formulation of E ̵ h ( r, S ) , as in Theorem 6.1, and that ̃ a ∈ C b ( R d ) and ̃ B ∈ L ( H ) . That is to say that ̃ a ( q, p ) + ̃ B ≤ ( Z − z ) and E ̵ h ( r, S ) = ∫ R d ̃ a ( z ) r ( z ) dz + trace L ( R d ) ( ̃ BS ) . Here we have used the notation z = ( q, p ) , dz = dqdp , and Z = ( Q, P ) .Let us denote by Π ( z ) an optimal coupling of r, S and let us define a ( z ) ∶ = (∣ z ∣ − ̃ a ( z )) B ∶ = (∣ Z ∣ − ̃ B ) . One has ( a ( z ) + B − z ⋅ Z ) ≥ L ( R d ) ∫ R d Π ( z ) ( a ( z ) + B − z ⋅ Z ) Π ( z ) dz = , Therefore, since Π ( z ) ( a ( z ) + B − z ⋅ Z ) Π ( z ) ≥ ( z ) ( a ( z ) + B − z ⋅ Z ) Π ( z ) = , . In other words,Π ( z ) ( a ( z ) + B − z ⋅ Z ) ( Π ( z ) ( a ( z ) + B − z ⋅ Z ) ) ∗ = ( a ( z ) + B − z ⋅ Z ) Π ( z ) = ( a ( z ) + B − z ⋅ Z ) Π ( z ) = . Hence, the range of Π consists in functions R d ∋ z ↦ ψ z ∈ L ( R d ) such that(14) ( a ( z ) + B − z ⋅ Z ) ψ z = ⇐⇒ ( B − z ⋅ Z ) ψ z = − a ( z ) ψ z ∶ UANTUM AND SEMIQUANTUM PSEUDOMETRICS 31 the vectors ψ z are the eigenvectors of B − z ⋅ Z with eigenvalue − a ( z ) . But B + a ( z ) − z ⋅ Z ≥
0. Therefore − a ( z ) is the lowest eigenvalue of B − z ⋅ Z . From now on, we will suppose that the fundamental of B − z ⋅ Z is non degenerate.This means that Π ( z ) is proportional to ∣ ψ z ⟩⟨ ψ z ∣ and therefore, since Π ( z ) is acoupling between r and S , Π ( z ) = r ( z )∣ ψ z ⟩⟨ ψ z ∣ and S = ∫ R d r ( z )∣ ψ z ⟩⟨ ψ z ∣ dz. We just prove the following result.
Theorem 9.1.
Let B be a bounded optimal Kantorovich operator of E ̵ h ( r, S ) . Letmoreover, for each z ∈ R d , ψ z be the ground state of B − z ⋅ Z .Then S admits the following representation S = ∫ R d r ( z )∣ ψ z ⟩⟨ ψ z ∣ dz. Theorem 9.1 suggests to associate to any probability density µ the followingoperator(15) µ Ð→ OP r,S ̵ h [ µ ] ∶ = ∫ R d ∣ ψ z ⟩⟨ ψ z ∣ µ ( dz ) . The arrow in (15) can be seen as the “optimal transport”, from classical proba-bility densities to quantum density matrices, transporting r to S .Note that, for any density µ ,trace OP r,S ̵ h [ µ ] = ∫ R d µ ( dz ) . Finally, using (13), we easily show, by analogy with the proof of Theorem 2.6(b) in [2], that, when a ∈ C ( R d ) , ( ∇ a ) r ∈ C b ( R d ) and, e.g., ψ z ∈ Dom ( i ̵ h [ Z, B ]) for all z ∈ supp ( r ) ,0 = Π ( z ) i ̵ h [ Z, ( a ( z ) + B − z ⋅ Z ) Π ( z )] = Π z i ̵ h [ Z, a ( z ) + B − z ⋅ Z ] Π ( z ) = Π ( z )([ Z, B ] − z ) Π ( z ) and0 = Π ( z )∣ Z, ( a ( z ) + B − z ⋅ Z ) Π ( z )} = Π ( z ){ Z, a ( z ) + B − z ⋅ Z ∣ Π ( z ) = Π ( z )( ∇ a ( z ) − Z ) Π ( z ) . Therefore the (classical and quantum) “gradient” aspect appears in the followingexpressions ⟨ ψ z ∣ Z ∣ ψ z ⟩ = ∇ a ( z ) z = ⟨ ψ z ∣ Z ∣ ∇ Q Bψ z ⟩ where ∇ Q ∶ = i ̵ h [ JZ, ⋅ ] with J the symplectic matrix defined by { f, g } = ∇ f ⋅ J ∇ g ,as introduced and motivated in [2, Section 1]. Let us finish this section by an example. Suppose that S = OP T ̵ h (( π ̵ h ) d r ) . In this case, one knows, [7, Theorem 2.4 (1)] (note a difference of normalization: in[7], E ̵ h = E ̵ h ), E ̵ h ( r, S ) = d ̵ h = ∫ R d ̃ a ( z ) r ( z ) dz + trace ( ̃ BS ) with ̃ a = , ̃ B = d ̵ hI H . Since ( q − x ) + ( p + i ̵ h ∇ x ) ≥ d ̵ hI H = ̃ a ( z ) I H + ̃ B , ̃ a and ̃ B are optimal and a ( q, p ) = ∣ z ∣ and B = (∣ Z ∣ − d ̵ h ) . Hence a ( z ) + B − z ⋅ Z = ( −∇ x + x − ( q + ip ))( ∇ x + x − ( q − ip )) , the solution of (14) is ψ z = ( π ̵ h ) − d / e − ( x − q ) ̵ h e i p.x ̵ h and Theorem 9.1 expresses back that S = OP T ̵ h (( π ̵ h ) d r ) andOP µ, OP T ̵ h ( µ )̵ h = OP T ̵ h for any probability density µ .9.2. A semiquantum Legendre transform.
As we have seen, − a ( z ) is the fun-damental of the operator B − z ⋅ Z . Therefore, by the variational characterizationof the lowest eigenvalue, − a ( z ) = inf φ ∈ Dom ( B )∥ φ ∥ H = (⟨ φ ∣ B ∣ φ ⟩ − z ⋅ ⟨ φ ∣ Z ∣ φ ⟩) , to be faced to the classical definition of the Legendre transform a ( z ) = sup z ′ ( z ⋅ z ′ − b ( z ′ )) . Let us define the semiquantum Legendre transform by B sq ∗ ∶ = sup φ ∈ Dom ( B )∥ φ ∥ H = ( z ⋅ ⟨ φ ∣ Z ∣ φ ⟩ − ⟨ φ ∣ B ∣ φ ⟩) . Theorem 9.2.
Let a ( z ) = (∣ z ∣ − ̃ a ( z )) , B = (∣ Z ∣ − ̃ B ) where ̃ a ( z ) and ̃ B arebounded optimal Kantorovich potentials for E ̵ h ( r, S ) . Then a = B sq ∗ . Proof.
We just recall the variational argument.Let A ≥ A ∣ φ ⟩ =
0. Then, ⟨ φ ∣ A ∣ φ ⟩ ≤ inf φ ∈ Dom ( B )∥ φ ∥ H = ⟨ φ ∣ A ∣ φ ⟩ and ⟨ φ + δφ ∣ A ∣ φ + δφ ⟩ = ⟨ δφ ∣ A ∣ δφ ) . (cid:3) UANTUM AND SEMIQUANTUM PSEUDOMETRICS 33
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